Ch. 34 - The Wave Nature of Light: Interference and Polarization

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 34 - The Wave Nature of Light: Interference and Polarization

Problem 23

Chapter 33, Problem 23

Suppose that one slit of a double-slit apparatus is wider than the other so that the intensity of light passing through it is twice as great. Determine the intensity I as a function of position (θ) on the screen for coherent light.

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Start by recalling the formula for the intensity of light in a double-slit interference pattern: \( I(\theta) = I_0 \cos^2(\phi/2) \), where \( \phi \) is the phase difference between the two waves arriving at a point on the screen.

Since one slit is wider and allows twice the intensity of light compared to the other, the amplitudes of the waves from the two slits will differ. Let the amplitude of the wave from the narrower slit be \( A \), and the amplitude from the wider slit will be \( \sqrt{2}A \) (since intensity is proportional to the square of the amplitude).

The total amplitude at a point on the screen is the vector sum of the two amplitudes. Using trigonometry, the resultant amplitude \( A_{\text{total}} \) is given by: \( A_{\text{total}} = \sqrt{A^2 + (\sqrt{2}A)^2 + 2A(\sqrt{2}A)\cos(\phi)} \).

Simplify the expression for \( A_{\text{total}} \): \( A_{\text{total}} = A\sqrt{1 + 2 + 2\sqrt{2}\cos(\phi)} = A\sqrt{3 + 2\sqrt{2}\cos(\phi)} \).

Finally, the intensity \( I(\theta) \) is proportional to the square of the total amplitude: \( I(\theta) = I_0 \left( 3 + 2\sqrt{2}\cos(\phi) \right) \), where \( \phi = \frac{2\pi d \sin(\theta)}{\lambda} \), \( d \) is the slit separation, and \( \lambda \) is the wavelength of the light.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Double-Slit Experiment

The double-slit experiment demonstrates the wave nature of light through interference patterns created when coherent light passes through two closely spaced slits. When light waves from the slits overlap, they can constructively or destructively interfere, resulting in bright and dark fringes on a screen. Understanding this setup is crucial for analyzing how varying slit widths affect intensity distribution.

Intensity of Light

Intensity refers to the power per unit area carried by a wave, which in the context of light, is related to the amplitude of the light wave. In the double-slit experiment, the intensity at a point on the screen is influenced by the contributions from both slits, with wider slits allowing more light to pass through, thus affecting the overall intensity pattern observed.

Interference Pattern

An interference pattern is formed when two or more coherent light waves overlap, leading to regions of constructive interference (bright spots) and destructive interference (dark spots). The intensity at any point on the screen can be calculated using the principle of superposition, taking into account the amplitude contributions from each slit, which is particularly important when one slit is wider than the other.

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Wave Interference & Superposition

Related Practice

Textbook Question

A uniform thin film of alcohol (n = 1.36) lies on a flat glass plate (n = 1.56). When monochromatic light, whose wavelength can be changed, is incident normally, the reflected light is a minimum for λ = 492 nm and a maximum for λ = 615 nm. What is the minimum thickness of the film?

Textbook Question

(II) Suppose a thin piece of glass is placed in front of the lower slit in Fig. 34–7 so that the two waves enter the slits 180° out of phase (Fig. 34–44). Describe in detail the interference pattern on the screen.

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Textbook Question

(II) Light of wavelength λ passes through a pair of slits separated by 0.17 mm, forming a double-slit interference pattern on a screen located a distance 35 cm away. Suppose that the image in Fig. 34–9a is an actual-size reproduction of this interference pattern. Use a ruler to measure a pertinent distance on this image; then utilize this measured value to determine λ (nm) .

Textbook Question

(I) If a soap bubble is 120 nm thick, what wavelength is most strongly reflected at the center of the outer surface when illuminated normally by white light? Assume that n = 1.35.

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Textbook Question

Consider three equally spaced and equal-intensity coherent sources of light (such as adding a third slit to the two slits of Fig. 34–12). Determine the positions of minima.

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Textbook Question

In a two-slit interference experiment, the path length to a certain point P on the screen differs for one slit in comparison with the other by 1.25λ.

(a) What is the phase difference between the two waves arriving at point P?

(b) Determine the intensity at P, expressed as a fraction of the maximum intensity Iₒ on the screen.